Problem: Rewrite as a simplified fraction. ${1.13\overline{1} = {?}}$
Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 1131.1111...\\ 100x &= 113.1111...\end{align*}}$ Subtracting the two equations, we have: ${900x = 1018}$ ${x = \dfrac{1018}{900}} $ ${x = \dfrac{509}{450}} \text{or} {1 \dfrac{59}{450}}$